Exercises From Cartan Subalgebras and Root Systems

  1. Why are diagonal matrices their own normaliser?
  2. How you do define determinant for a general linear operator? (i.e without defining a basis)
  3. Why is the exponential map well defined for matrices? exp: gl_{n} \rightarrow GL(g) Where the exponential map takes a matrix to its exponential Taylor series.
  4. Consider gl_{2} \& gl_{3}, what is \mathrm{ad}(x), and what is e^{\mathrm{ad}(x)} ?
  5. Show the rank of  gl_{n} is n, where rank is the dimension of a Cartan Subalgebra.
  6. Show that the regular elements in gl_{n} are diagonal matrices with no repeated entries.
  7. If we define the map -1 : V \rightarrow V which takes a root \alpha \rightarrow -\alpha, For which root systems is this in the Weyl Group?
  8. (Very optional) Why is the Dynkin diagram for E_{8} not allowed to have a connection on the central root?

Don’t forget that next week is the problem session!

Future talk Schedule

Hey guys,

Sorry this is coming through a bit late, been a crazy week!

The talk schedule which was decided on Wednesday afternoon is as follows:

  1. Reuben, Intro to Lie algebra, nilpotent and solvable (Serre 1, FH 9.1,9.2)
  2. Chris, Semi-simple Lie algebras (Serre 2, FH 9.3,9.4)
  3. Pat, Lie algebras of dim 1,2,3 (FH 10)
  4. Stephen, Rep theory of sl(2,C) (Serre 4, FH 11)
  5. Reuben, Cartan Subalgebras (Serre 3, FH 14 & App. D)
  6. Mitchell, Root Systems I (Serre 5, FH 21)
  7. Jackson, Structure of semi-simple Lie algebras
  8. Alex, Reps of semi-simple Lie algebras

That covers the talks that should take us to the end of the course. Other things that were mentioned on Wednesday were that the midterm would assess questions on reps of finite groups, and then the final project would be about Lie algebras.

(edit 26/3: Swapped Reuben and Alex)